Censored regression, selection model, weak IV, and quantile regression

class: center, middle, inverse, title-slide

# Censored regression, selection model, weak IV, and quantile regression
## Tutorial 1
### Stanislav Avdeev

---

# Tutorials

- 7 TA sessions
  - 6 TA sessions are about lecture material
  - The last session is primarily about exam and remaining questions about the course material (TBA)
- Send me **any** questions you want to discuss before each TA session
  - Use Canvas or send me an email (s.avdeev@tinbergen.nl)
  - Alternately, leave your questions anonymously here: https://onlinequestions.org (enter the event code 18631)

---

# Assignments

- Due date: 11:59pm on Sundays (the first assignment is an exception: 11:59am on Tuesday)
- Assignments are graded within a week from the deadline
- Solutions will not be shared so if you want to discuss a specific exercise, let me know before the TA session (you submit your solutions on Sunday, thus, we can discuss any questions on the following TA session on Tuesday)

---

# Course objective
- The key objective of the course is **applying** microeconometric techniques rather than **deriving** econometric and statistical properties of estimators
- In other words, there’s way less of this:

`\begin{align*}
  \text{plim} \hat{\beta}_{OLS} = \beta + \text{plim} (\frac{1}{N}X'X)^{-1} \text{plim} \frac{1}{N} X' \varepsilon = \beta + Q^{-1} \text{plim} \frac{1}{N} X' \varepsilon
\end{align*}`

- And way more of this:

```r
library(fixest)

df <- tibble(groups = sort(rep(1:10, 600)), 
             time = rep(sort(rep(1:6, 100)), 10),
             Treated = I(groups > 5) * I(time > 3),
             Y = groups + time + Treated * 5 + rnorm(6000))
did <- feols(Y ~ Treated | groups + time, data = df)
```

- If you would like to go deeper into the former, take Advanced Econometrics I and II next year

---

# Goal for today's tutorial

1. Use a tobit model to estimate censored data
1. Discuss a sample selection model and implement the selection mechanism
1. Work with strong and weak instrumental variables
1. Test instrumental variables
1. Work with a quantile regression and discuss inference tools

---

# Censored regression

- Censoring occurs when the value of a variable is limited due to some constraint
- For example, we tend not to see some values of self declared earnings with discrete categories (if earn **at least** `$3500$` euro per month, write `$3500$`)
- In this case OLS estimates are biased
- A standard method to account for censoring is to combine a probit model with OLS, i.e. tobit model

---

# Censored regression: simulation

- The clearest way to understand how a certain estimator works is to generate data yourself so you know the true **data generating process** - DGP
- Let's estimate returns to education: does education increase wages?
- But suppose that we do not observe wages below a specific threshold (due to the features of a questionnaire, privacy concerns, coding, etc.)
- We need to generate data containing years of education and wages

---

# Censored regression: simulation

```r
# Always set seed so you can replicate your results
set.seed(7)
df <- tibble(education = runif(1000, 5, 15),
             wage_star = 1000 + 200*education + rnorm(1000, 0, 100),
             wage = ifelse(wage_star > 3500, 3500, wage_star))  %>%
  arrange(desc(wage_star))
```

```
## # A tibble: 3 × 3
##   education wage_star  wage
##       <dbl>     <dbl> <dbl>
## 1      14.9     4202.  3500
## 2      14.8     4174.  3500
## 3      14.9     4167.  3500
```

```
## # A tibble: 3 × 3
##   education wage_star  wage
##       <dbl>     <dbl> <dbl>
## 1      5.26     1890. 1890.
## 2      5.04     1878. 1878.
## 3      5.06     1837. 1837.
```

---

# Censored regression: OLS

- Now let's pretend that we do not know the DGP and simply apply OLS

```r
ols_model <- lm(wage ~ education, df)
```

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Model 1 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 1255.542*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (14.018) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> education </td>
   <td style="text-align:center;"> 167.686*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (1.340) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

- Using these OLS estimates, we would wrongly conclude that "an additional year of education is associated with `$167.686$` increase in monthly wages"

---

# Censored regression: tobit-model

- But these are biased estimates since we know the true effect is `$200$` (remember DGP)
- Let's try to recover unbiased effects of education on wages by using tobit-model
- The solution provided by the tobit-model is to
  - use a probit model to account for the censoring 
  - estimate OLS on the non-censored data
- Tobit-model estimator is easy to implement with `censReg` package

---

# Censored regression: tobit-model

- Remember that we have right censored wages: wages above `$3500$` are coded as `$3500$`

```r
library(censReg)
tobit_model <- censReg(wage ~ education, data = df, right = 3500)
```

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Model 1 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 1011.629*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (13.998) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> education </td>
   <td style="text-align:center;"> 198.053*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (1.495) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> logSigma </td>
   <td style="text-align:center;"> 4.586*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (0.026) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

- We recovered the unbiased estimates of returns to education

---

# Censored regression: graphically

- We will use a lot of graphs since they provide more intuition of what is happening

![](data:image/png;base64,#tutorial1_files/figure-html/unnamed-chunk-8-1.png)

---

# Censored regression: some remarks

- You can specify both left and right censoring using `censReg` function
- Important assumption of the tobit-model is that the unobserved term is normally distributed (which is the case in our simulated dataset)
- If the data is missing not because the outcome variable is **above (or below)** some threshold but because individuals in the data have made a **choice** such that we can't observe their outcome variable, we can't use censoring
- Censoring cannot be applied because the availability of data is influenced by the choice of agents (i.e. selection on unobservables)
- It is a typical sample selection problem

---

# Sample selection model

- Let us consider the case of studying female’s wages
- Usually, wages are observed for a fraction of women in the sample, whereas the remaining part of women are observed as unemployed or inactive
- If we run OLS regression using the observed wages, this would deliver consistent estimations only if working females are a random sample of the population
- However, theory of labor supply suggests that this may not be the case, since (typically) female labor supply is sensitive to household decisions
- That is, female workers self-select into employment, and the self-selection is not random
- This difference may lead us to underestimate the gender wage gap

---

# Sample selection model

- Suppose a female worker decides to work or not based on a latent variable `$I_i^*$` (say, utility derived from working), which depends on a set of observed `$Z_i$` and unobserved `$V_i$` characteristics

`\begin{align*}
  I_i^* = Z_i ' \gamma + V_i
\end{align*}`

- The indicator function (decision to work or not), based on `$I_i^*$`, takes two values

`\begin{align*}
  I_i = \begin{cases} \mbox{} 1 \text{ (working) } \ & \mbox{} \text{ if } I_i^* > 0 \\ \mbox{} 0 \text{ (not working) } & \mbox{} \text{ if } I_i^* \leq 0 \end{cases}
\end{align*}`

- Suppose there is a latent outcome `$Y_i^*$`, i.e. wages of female workers, which depend on a set of observed `$X_i$` and unobserved `$U_i$` characteristics

`\begin{align*}
  Y_i^* = X_i ' \beta + U_i
\end{align*}`

- However, we observe wages only for females who decide to work. `$Y_i$` are observed wages that equal to

`\begin{align*}
  Y_i = \begin{cases} \mbox{} Y_i^* \ & \mbox{} \text{ if } I_i = 1 \\ \mbox{} \text{missing} & \mbox{} \text{ if } I_i = 0 \end{cases}
\end{align*}`

---

# Sample selection model: assumptions

- To estimate the sample selection model, distributional assumptions on the disturbances terms are made, such as bivariate normality

`\begin{align*}
  \left[\begin{array}{l}
    U_{i} \\
    V_{i}
  \end{array}\right] \sim \mathcal{N}\left(0,\left[\begin{array}{cc}
    \sigma^{2} & \rho \sigma \\
    \rho \sigma & 1
  \end{array}\right]\right)
\end{align*}`

- Note that the variance of the normal distribution is not identified in the probit model so it is set to 1

---

# Sample selection model: simulation

- Let's simulate a dataset with the selection mechanism

```r
library(mvtnorm) # to simulate bivariate normal random variable
set.seed(7)
df <- tibble(z = runif(1000),
             x = runif(1000),
             uv = rmvnorm(1000, mean = c(0, 0), 
                  sigma = rbind(c(1, 0.7), c(0.7, 1))),
             i_star = 4 - 5 * z + uv[, 1],
             y_star = 6 - 3 * x + uv[, 2],
             # this is our selection mechanism
             y = ifelse(i_star > 0, y_star, 0))
head(df)
```

```
## # A tibble: 6 × 6
##        z     x  uv[,1]   [,2]  i_star y_star     y
##    <dbl> <dbl>   <dbl>  <dbl>   <dbl>  <dbl> <dbl>
## 1 0.989  0.418  0.865  -0.237 -0.0799   4.51  0   
## 2 0.398  0.813  0.0624 -0.256  2.07     3.31  3.31
## 3 0.116  0.278 -0.540   0.229  2.88     5.39  5.39
## 4 0.0697 0.968 -1.74   -1.90   1.91     1.20  1.20
## 5 0.244  0.247 -0.250   0.594  2.53     5.85  5.85
## 6 0.792  0.905 -1.78   -0.733 -1.74     2.55  0
```

---

# Sample selection model: simulation

- The true effect of `$Z$` on `$I$` (decision to work) is `$-5$` and the effect `$X$` on `$Y$` (wages) is `$-3$`

```r
selection_equation <- glm(I(y > 0) ~ z, df, family = binomial(link = "probit"))
wage_equation <- lm(y ~ x, df)
```

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Model 1 </th>
   <th style="text-align:center;"> Model 2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 4.457*** </td>
   <td style="text-align:center;"> 4.635*** </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.287) </td>
   <td style="text-align:center;"> (0.135) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> z </td>
   <td style="text-align:center;"> −5.647*** </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.379) </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> −2.074*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (0.237) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 1000 </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

- Clearly, the estimates are biased since `$\text{cov} (U_i, V_i) \neq 0$`

---

# Sample selection model: Heckman

- To solve the sample selection problem, one needs to use the **Heckman selection model** (left as an exercise in the `$1^{\text{st}}$` assignment)
- Heckman estimator is very similar to the Tobit estimator 
- The difference is that this estimator allows for a set of characteristics that determine whether or not the outcome variable is censored

---

# IV

- The basic idea of the IV estimator is to
  - use the instrument to predict treatment
  - use that predicted treatment to predict the outcome
- We need a separate equation for each of those steps
  - **First stage**: predict treatment `$X_1$` with the instrument `$Z$`, with a control variable `$X_2$`

`\begin{align*}
  X_1 = \gamma_0 + \gamma_1Z + \gamma_2X_2 + V
\end{align*}`

- **Second stage**: use that equation to predict `$X_1$`, getting `$\hat{X_1}$`. Then, use those predictions to predict `$Y$`

`\begin{align*}
  Y = \beta_0 + \beta_1\hat{X_1} + \beta_2X_2 + U
\end{align*}`

- Notice that we need to use all exogenous variables in both stages

---

# IV: conditions

- For the IV to work, we need two things to hold
  - **Relevance**: the instrument must be a *strong predictor* of the treatment. It can't be trivial or unimportant
`\begin{align*}
  \text{cov}(X, Z) \neq 0
\end{align*}`
  - **Validity**: the instrument must actually be *exogenous* (or at least exogenous after adding controls)
`\begin{align*}
  \text{cov}(Z, U) = 0
\end{align*}`

---

# IV: small-sample bias

- IV is actually a *biased* estimator
- The mean of its sampling distribution is *not* the population parameter
- It *would be* the population parameter at infinite sample size, but we don't have that
- In small samples, the bias of IV is

`\begin{align*}
  \frac{\text{cov} (Z, U)}{\text{cov} (X, Z)}
\end{align*}`

- If `$Z$` is valid, then in infinite samples `$\text{cov} (Z, U) = 0$` and this goes away. But in a non-infinite sample, it will be nonzero by chance, inducing some bias. The smaller the sample, the more likely we are to get a large value by random chance
- The bias is smaller 
  - the stronger the relationship between `$X$` and `$Z$`
  - the smaller the sum of squared errors
  - the bigger the variation in `$X$`
- What happens when `$\text{cov}(X, Z)$` is small?

---

# IV: weak instrument problem

- If `$Z$` has only a trivial effect on `$X$`, then it's not **relevant** (even if it's truly **exogenous**)
- And our **small-sample bias** will be big
- Thus, weak instrument problem means that we probably shouldn't be using IV in small samples
- This also means that it's really important that `$\text{cov} (X, Z)$` is not small
- There are some rules of thumb for how strong an instrument must be to be counted as "not weak"
- A t-statistic above `$3$`, or an F statistic from a joint test of the instruments that is `$10$` or above
- These rules of thumb aren't great - selecting a model on the basis of significance naturally biases your results
- What you really want is to know the **population** effect of `$Z$` on `$X$` - you want the F-statistic from that to be bigger than `$10$`. Of course we don't actually know that

---

# Weak IV: estimation

- There are a bunch of ways to do the IV analysis
  - the classic one is `ivreg` in the `AER` package, 
- Other functions are more fully-featured, including robust SEs, clustering, and fixed effects
  - `feols` in `fixest`
  - `felm` in `lfe`
  - `tsls` in `sem`
  - `ivpack`

- We'll be using `feols` from `fixest`

---

# Weak IV: simulation

- Let's create a dataset with instruments

```r
library(fixest)
set.seed(7)
df <- tibble(z1 = rnorm(1000),
             u1 = rnorm(1000),
             # x1 is endogenous since it correlates with u1 by construction
             x1 = 0.2*z1 + 4*u1 + rnorm(1000),
             y1 = 3*x1 + 5*u1)
head(df)
```

```
## # A tibble: 6 × 4
##       z1     u1     x1        y1
##    <dbl>  <dbl>  <dbl>     <dbl>
## 1  2.29   1.25   6.86   26.8    
## 2 -1.20  -0.766 -5.40  -20.0    
## 3 -0.694  0.216 -0.358   0.00737
## 4 -0.412 -0.364 -1.91   -7.55   
## 5 -0.971 -0.821 -2.72  -12.3    
## 6 -0.947  0.582  2.32    9.87
```

---

# Weak IV: simulation

```r
# The true effect is 3
library(fixest)
ols_model <- lm(y1 ~ x1, df)
iv_model <- feols(y1 ~ 1 | x1 ~ z1, df, se = 'hetero')
```

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Model 1 </th>
   <th style="text-align:center;"> Model 2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> x1 </td>
   <td style="text-align:center;"> 4.171*** </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.009) </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> fit_x1 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 3.479*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (0.348) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 1000 </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

- `$Z_1$` is a pretty effective instrument even if the correlation between `$Z_1$` and `$X_1$` is small
  - Check relevance: `$\text{cov} (X_1, Z_1) =$`  0.068, so it's a weak instrument
  - Check validity: `$\text{cov} (Z_1, U_1) =$` 0.027, pretty close to zero

---

# Weak IV: simulation

- Remember that usually we can't test the **validity** assumption when we have one instrument, but we know the DGP in this case
- Now let's see what happens when there is a small correlation between `$Z$` and `$U$`
- Imagine there is some additional explanatory variable `$V$` which is unobserved but partially explains the instrument

```r
set.seed(7)
df <- tibble(v = rnorm(1000),
             z2 = -v + rnorm(1000),
             u2 = 0.1*v + rnorm(1000),
             # all coefficients stay the same
             x2 = 0.2*z2 + 4*u2 + rnorm(1000),
             y2 = 3*x2 + 5*u2)
head(df)
```

```
## # A tibble: 6 × 5
##        v      z2      u2     x2     y2
##    <dbl>   <dbl>   <dbl>  <dbl>  <dbl>
## 1  2.29  -1.04    1.64    7.57   30.9 
## 2 -1.20   0.431  -2.22   -9.87  -40.7 
## 3 -0.694  0.910  -1.15   -5.12  -21.1 
## 4 -0.412  0.0479 -0.412  -3.49  -12.5 
## 5 -0.971  0.150   0.659   4.85   17.9 
## 6 -0.947  1.53    0.0833  0.908   3.14
```

---

# Weak IV: simulation

```r
# The true effect is 3
iv_model2 <- feols(y2 ~ 1 | x2 ~ z2, data = df, se = 'hetero')
ols_model2 <- lm(y2 ~ x2, df)
```

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Model 1 </th>
   <th style="text-align:center;"> Model 2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> x2 </td>
   <td style="text-align:center;"> 4.163*** </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.009) </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> fit_x2 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.944 </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (4.365) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 1000 </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

- In this case we get a better estimate by using the biased OLS estimator than by using IV 
- Why? Because of the weak instrument problem
  - Check relevance: `$\text{cov} (X_2, Z_2) =$` 0.021
  - Check validity: `$\text{cov} (Z_2, U_2) =$` -0.036

---

# Weak IV: simulation

- These results are primarily a function of the weakness of `$Z$` at explaining `$X$`. Let's see what happens if `$Z$` has more explanatory power

```r
set.seed(7)
df <- tibble(v = rnorm(1000),
             z2 = -v + rnorm(1000),
             u2 = 0.1*v + rnorm(1000),
             # we only change the coefficient for z2 in the equation for x3
             x3 = 3*z2 + 4*u2 + rnorm(1000),
             y2 = 3*x3 + 5*u2)
head(df)
```

```
## # A tibble: 6 × 5
##        v      z2      u2    x3    y2
##    <dbl>   <dbl>   <dbl> <dbl> <dbl>
## 1  2.29  -1.04    1.64    4.65  22.2
## 2 -1.20   0.431  -2.22   -8.66 -37.1
## 3 -0.694  0.910  -1.15   -2.57 -13.5
## 4 -0.412  0.0479 -0.412  -3.36 -12.1
## 5 -0.971  0.150   0.659   5.27  19.1
## 6 -0.947  1.53    0.0833  5.19  16.0
```

---

# Weak IV: simulation

```r
# The true effect is 3
iv_model3 <- feols(y2 ~ 1 | x3 ~ z2, data = df, se = 'hetero')
ols_model3 <- lm(y2 ~ x3, df)
```

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Model 1 </th>
   <th style="text-align:center;"> Model 2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> x3 </td>
   <td style="text-align:center;"> 3.578*** </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.020) </td>
   <td style="text-align:center;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> fit_x3 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 2.956*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (0.036) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 1000 </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

- Even though the correlation between `$Z$` and `$U$` is the same as previously, the strength of the instrument in explaining `$X$` wins out and gives us a better estimator than OLS
  - Check relevance: `$\text{cov} (X_3, Z_2) =$` 0.697
  - Check validity: `$\text{cov} (Z_2, U_2) =$` -0.036

---

# Weak IV: F-test

- Let's look at the F-test from the output of `feols()` using a simulated dataset

```r
thef <- fitstat(iv_model3, 'ivf', verbose = FALSE)$`ivf1::x3`$stat
```

- 941.48 is way above 10
- Lee, D., et al. (2021) discuss the potentially severe large-sample distortions from using conventional value of the F-test `$10$` and they suggest to use as a rule of thumb the minimum value of the F-test `$104.7$`, which is needed to ensure a test with significance level `$0.05$`

---

# Weak IV: overidentification test

- "Overidentification" just means we have more identifying conditions (**validity** assumptions) than we actually need. We only need one instrument, but we have two (or more)
- So we can compare what we get using each instrument individually
- If we assume that **at least one of them is valid**, and they both produce similar results, then that's evidence that **both** are valid

---

# Weak IV: simulation

- We can do this using `diagnostics = TRUE` in `iv_robust` again

```r
set.seed(7)
# Create data where z1 is valid and z2 is invalid
df <- tibble(z1 = rnorm(1000), 
             z2 = rnorm(1000),
             x = z1 + z2 + rnorm(1000),
             y = 2*x + z2 + rnorm(1000))

iv <- feols(y ~ 1 | x ~ z1 + z2, df, se = 'hetero')
fitstat(iv, 'sargan')
```

```
## Sargan: stat = 248.7, p < 2.2e-16, on 1 DoF.
```

- The null hypothesis of the Sargan test is that the covariance between the instrument and the error term is zero
`\begin{align*}
  \text{cov } (Z, U) = 0
\end{align*}`
- Thus, rejecting the null indicates that at least one of the instruments is not valid
- So we reject the null, indicating that one of the instruments is endogenous (although without seeing the true DGP we couldn't guess if it were `$Z_1$` or `$Z_2$`)

---

# Weak IV: simulation

- The true effect is `$2$`

```r
iv1 <- feols(y ~ 1 | x ~ z1, df, se = 'hetero')
iv2 <- feols(y ~ 1 | x ~ z2, df, se = 'hetero')
```

<table style="NAborder-bottom: 0; width: auto !important; margin-left: auto; margin-right: auto;" class="table">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Model 1 </th>
   <th style="text-align:center;"> Model 2 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 0.029 </td>
   <td style="text-align:center;"> 0.012 </td>
  </tr>
  <tr>
   <td style="text-align:left;">  </td>
   <td style="text-align:center;"> (0.043) </td>
   <td style="text-align:center;"> (0.049) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> fit_x </td>
   <td style="text-align:center;"> 2.058*** </td>
   <td style="text-align:center;"> 2.912*** </td>
  </tr>
  <tr>
   <td style="text-align:left;box-shadow: 0px 1px">  </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (0.044) </td>
   <td style="text-align:center;box-shadow: 0px 1px"> (0.046) </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Num.Obs. </td>
   <td style="text-align:center;"> 1000 </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
</tbody>
<tfoot><tr><td style="padding: 0; " colspan="100%">
<sup></sup> + p &lt; 0.1, * p &lt; 0.05, ** p &lt; 0.01, *** p &lt; 0.001</td></tr></tfoot>
</table>

---

# Quantile regression

- Consider the very simple OLS version testing this model using the experimental data:
`\begin{align*}
  Y_i = \alpha + D_i ' \beta + U_i
\end{align*}`
where `$Y_i$` is an outcome variable, `$D_i$` is a tretment variable
- Recall that this will estimate our ATE for the treatment
- What is the interpretation of this affect?
  - `$E(Y_i (1)) - E(Y_i (0))$`, i.e. the expected change in the outcome for a person moving from untreated to treated. That’s a useful metric
- In other words, it characterizes the **mean** of our outcome variable

---

# Quantile regression

- What if we care about other things but the mean?
  - What are the factors influencing total medical expenditures for people with low-, medium-, and high- expenditures?
  - What are the effects of a training program on employment opportunities for people with the different number of years of education?
- Quantile regression also solves the problems with
  - Skewed variables – no more worrying about logs or outliers in the outcome variable
  - Censoring – in many datasets, our outcome variables are top-coded or bottom-coded
- But it has its own issues
  - it is noisier
  - it is challenging to interpret in an intuitive way
- If you have underlying theory that has implications for distribution, quantile regression is the rigth tool for empirical analysis

---